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Delay-dependent stability of reset systems
Automatica 46 (2010) 216–221
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Brief paper
Delay-dependent stability of reset systemsI Antonio Barreiro a,∗ , Alfonso Baños b a
E.T.S. Ingenieros Industriales, University of Vigo, 36200 Vigo, Spain
b
Facultad de Informática, University of Murcia, 30071 Murcia, Spain
article
info
Article history: Received 9 July 2008 Received in revised form 23 July 2009 Accepted 21 October 2009 Available online 18 November 2009 Keywords: Reset control Hybrid systems Time-delay systems Passivity Lyapunov–Krasovskii functionals
abstract This work presents results on the stability of time-delay systems under reset control. The case of delaydependent stability is addressed, by developing a generalization of previous stability results for reset systems without delay, and also a generalization of the delay-independent case. The stability results are derived by using appropriate Lyapunov–Krasovskii functionals, obtaining LMI (Linear Matrix Inequality) conditions and showing connections with passivity and positive realness. The stability conditions guarantee that the reset action does not destabilize the base LTI (Linear Time Invariant) system. Several interpretations are given for these conditions in terms of impulsive control, which provide insights into the potentials of reset control. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The idea of reset control and in particular Clegg’s integrator (Clegg, 1958) and the first-order reset element (FORE) (Horowitz & Rosenbaum, 1975; Krishman & Horowitz, 1974) are attempts to achieve fast and robust control solutions for problems under linear limitations, which are particularly severe for plants with right-half plane poles or zeros, or with time delays (Åström, 2000). The implementation of reset control is very simple: it consists of resetting the state (or part of it) of a feedback LTI compensator (the ‘base’ LTI compensator) at every instant in which the tracking error is zero. A design approach (Horowitz & Rosenbaum, 1975; Krishman & Horowitz, 1974) is to tune the base LTI controller so that the base loop becomes stable, with fast transient and large bandwidth. Then, a reset action is included to achieve reduced overshoot and large stability margins. However, this method should be carefully applied since the reset may destabilize a well-designed LTI system. A solution to this problem has been reported in Beker, Hollot, Chait, and Han (2004) for finite-dimensional LTI plants. The result is a frequency domain condition, the Hβ condition, that guarantees closed loop stability of the reset system. Since then, a number of
I Work supported by MEC, Spain, under project DPI-2007 66455. The material in this paper was partially presented at the American Control Conference, New York, July 11–12, 2007. This paper was recommended for publication in revised form by Associate Editor Zhihua Qu under the direction of Editor Andrew R. Teel. ∗ Corresponding author. Tel.: +34 986 812232; fax: +34 986 814014. E-mail addresses: [email protected] (A. Barreiro), [email protected] (A. Baños).
0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2009.10.029
useful results have appeared for reset systems (Baños, Carrasco, & Barreiro, 2007; Hespanha, Liberzon, & Teel, 2008; Guo, Wang, Zheng, & Xie, 2007; Haddad, Nersesov, & Chellaboina, 2003; Nešić, Zaccarian, & Teel, 2005). However, none of them considered timedelay plants. As reset control is able to overcome fundamental limitations, and time delay is one source of such limitations, it is then of great interest to study the problem of delayed reset systems. It should be emphasized that this is extremely difficult: the combination of reset and delay moves the problem to the field of hybrid and distributed systems. Even the issue of existence and uniqueness of solutions may cause trouble unless special care is taken, for example, by means of the time-regularization procedure (Nešić et al., 2005). It should be noted that stability of impulsive systems is a very active research area. Two classical monographs are Bainov and Simeonov (1989) and Lakshmikantham, Bainov, and Simeonov (1989). In general, as reported in Bainov and Simeonov (1989), impulsive systems may be classified in (i) systems with impulses at fixed instants, (ii) systems with impulse effect at variable instants, an (iii) autonomous systems with impulse effects. Both (ii) and (iii) have state-dependent impulses, but there is an important difference: the reset surface is time independent in case (iii). Traditionally, most of the research effort has been dedicated to cases (i) and (ii), and thus case (iii), which includes reset control systems, is less developed. In addition, in the control field the main effort has been done for systems without time delays (Haddad et al., 2003; Yang, 2001). For the case of impulsive delay systems, available results in the literature are concentrated on case (i) (see De la Sen and Luo (2003), Liu and Ballinger (2001), Liu, Shen, Zhang,
A. Barreiro, A. Baños / Automatica 46 (2010) 216–221
and Wang (2007), and references therein), and case (ii) (see Liu and Wang (2006)). The objective of the present paper is to analyze the stability of time-delay systems under reset control. In particular, we try to find generalizations of the results in Beker et al. (2004) for reset systems without time delay, and also generalizations of the results in Baños and Barreiro (2009) for the delay-independent case. The generalization of these results suggests, for continuity, using the same common basic stability tool (Proposition 3.1 below), a theorem which requires that the energy (Lyapunov) must always decrease during the continuous and the impulsive behaviors. This basic result may be conservative, in general, and within the area of impulsive systems there is a vast amount of literature offering alternative techniques. At the same time, notice that many reset design approaches (Beker et al., 2004; Horowitz & Rosenbaum, 1975; Haddad et al., 2003; Krishman & Horowitz, 1974) are based on the useful idea of tuning the base system for a fast transient, underdamped but stable, and introducing reset for adding extra damping and decreasing overshoot. Within this idea, both the continuous and the impulsive mode must be stable, and so the basic tool (Proposition 3.1) is very well suited and adequate. Once the basic criterion (Proposition 3.1) is chosen, the motivation here is to explore its limits, in particular by obtaining generalizations of Baños and Barreiro (2009) and Beker et al. (2004). Following this, the original contributions of this paper are as follows. Firstly, the so-called Hβ condition in Beker et al. (2004) is extended to time-delay systems, as in Baños and Barreiro (2009), but now obtaining delay-dependent results, and with a more general structure C = (C1 , C2 , 0) of the output matrix (Proposition 3.2). Secondly, for passing from the time domain (Lyapunov and LMIs) to the frequency domain, an original idea is presented, based on interpreting resetting as an impulsive control around a certain base linear system (Propositions 3.3 and 3.4). This idea is different from the technique used in Baños and Barreiro (2009), based on the Kalman–Yacubovitch–Popov (KYP) lemma. Thirdly, the application of the impulse approach requires a careful treatment of the associated passivity conditions, as Dirac impulses are not L2 signals. This is analyzed in Proposition 3.5 and Remark 3. The organization of the paper is as follows. After introducing the problem in Section 2, the main stability results are given in Section 3. First, in 3.1, a Lyapunov–Krasovskii functional is used to derive delay-dependent stability conditions in the LMI form. Then, in 3.2, a second approach is developed based on passivity analysis. Section 4 presents an example and Section 5 gives our final conclusions. 2. Problem motivation and statement Consider a control system formed by a plant P0 and a reset controller K (both single-input–single-output):
P0 :
x˙ p = Ap xp + Bp up yp = Cp xp ,
(2.1)
K :
x˙ c = Ac xc + Bc uc xc ( t + ) = A ρ xc ( t ) y = C x + D u , c c c c c
(2.2)
where the second equation of the controller represents the impulsive or reset actions, applied when a certain reset condition, specified later, holds. If the connection from P0 to K (from K to P0 ) is affected by a delay h1 (h2 ), then it is easy to see that the total delay, h = h1 + h2 , can be moved to the plant input or output. For instance, move the delay to the plant input, as shown in Fig. 1. As the main interest is stability, let us suppose that the exogenous loop signals are zero (r = d = n = 0), so that the closed loop connections are up (t ) = yc (t − h),
uc (t ) = − yp (t ),
(2.3)
217
Fig. 1. Reset control system setup.
and the closed loop system is easily described by
x˙ p (t ) x˙ c (t )
x (t ) =A p xc (t )
x p ( t − h) , x c ( t − h)
+ Ad
(2.4)
and by impulsive dynamics xc (t + ) = Aρ xc (t ). With this in mind, we generalize to the system given by the impulsive-delayed differential equation x˙ (t ) = A x(t ) + Ad x(t − h), x(t + ) = AR x(t ),
IDDE
iff x(t ) 6∈ M , iff x(t ) ∈ M ,
(2.5)
for arbitrary A and Ad . It remains to introduce AR and M. Let us suppose that only the second part of the controller states are reset to zero; that is, Aρ = diag(1, . . . , 1, 0, . . . , 0). Then AR takes the form Beker et al. (2004) AR = diag In12 , 0n3 ,
(2.6)
so, the reset action acts over the last n3 states of the state x ∈ Rn . These dimensions are defined by n = n1 + n2 + n3 =
| {z } n23
n + n +n , | 1 {z }2 3
(2.7)
n12
where n1 (n23 ) is the number of states of the plant (controller). Accordingly, partition the state vector: > > x> = ( x> 1 , x2 , x3 ).
(2.8)
|{z} | {z } x> p
x> c
In principle, it is considered that the reset is applied at the time t when x(t ) hits the hypersurface M = x ∈ Rn : Cx = 0 ,
(2.9)
where the linear resetting condition has the structure 0 = Cx(t ) = C1 x1 (t ) + C2 x2 (t ).
(2.10)
As it is desired that 0 = Cx defines a hypersurface, then C is a row, C ∈ R1,n , and we assume here for C the structure C = (C1 , C2 , 0), which is more general than C = (C˜ 1 , 0, 0), the one adopted in Beker et al. (2004). For comparison with this last case, notice that there exists an invertible transformation T ∈ Rn,n such that C˜
T
C }| z }| { z }| { z { (C1 , C2 , 0) = (C˜ 1 , 0, 0) diag T12 , In3 .
(2.11)
One possible solution is C˜ 1 = (1, 0, . . . , 0) and T12 with its first row equal to (C1 , C2 ) and the remaining rows completed for full rank. In this way, (2.5), (2.6) and (2.9) define the system that we are going to study. It could be possible to adopt, following Nešić et al. (2005), the useful idea of time regularization, as a strategy for ensuring well-posedness and avoiding beating and Zenoness. However, this is not addressed here, for the sake of brevity and because the stability problem is essentially the same. 3. Stability analysis In this section, the stability of the delayed reset control system (2.5) is analyzed. Note that it is important to distinguish between the (lumped) state x(t ) ∈ Rn and the true (distributed) state xt : [−h, 0] → Rn , where xt (θ ) = x(t + θ ), θ ∈ [−h, 0]. In what follows, C ([−h, 0], Rn ) stands for the set of piecewise
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continuous functions xt (·) mapping [−h, 0] to Rn , with a finite number of discontinuity points. To address stability, consider certain Lyapunov–Krasovskii candidate functionals V (xt ), and use as a basic tool the following proposition (Beker et al., 2004). Proposition 3.1. Let V (xt ) : C ([−h, 0], Rn ) → R be continuously differentiable, positive-definite, radially unbounded, such that, along the solutions of (2.5), d dt
V (xt ) < 0,
xt 6= 0,
1 V (xt ) = V (xt + ) − V (xt ) ≤ 0,
x(t ) ∈ M .
3.1. Stability analysis based on LMIs
V (xt ) = V1 (xt ) + V2 (xt ) + V3 (xt ),
(3.1)
where the first term is V1 (xt ) =
>
−h >
1 2
x (t ) P x(t ), the second >
x (t + θ ) Q x(t + θ ) dθ , and V3 (xt ) = >
Ad XAd G xt +ξ dξ dθ , with G xt +ξ = Ax(t + −h θ G xt +ξ ξ ) + Ad x(t + ξ − h). Thus, by applying Proposition 3.1 to V (xt ),
we obtain sufficient conditions for global asymptotical stability. The requirement that d [V (xt )] /dt < 0 has been already treated in Park (1999), and it is finally expressed in LMI form. The requirement that 1V (xt ) ≤ 0 when x(t ) ∈ M can be addressed in a special form, related to the one in Theorem 5 in Beker et al. (2004). By combining these two approaches, we obtain the following proposition. Proposition 3.2. The system (2.5) is globally and asymptotically stable, if there exist Q > 0, V > 0, W , and P = P > = Pij i,j=1,2,3 > 0,
(3.2)
with
(0, 0, I ) P = Mβ (C1 , C2 ), P33 ,
(3.3)
for some matrix Mβ of adequate size, and such that
N
? ? ?
> >
−WAd
A Ad V
−Q ?
> >
0
Ad Ad V −V 0
h W+ 0 0 −V
{z
(3.5)
}
K
where I is the n3 × n3 identity and K1 is such that Imag K1 = Ker C˜ 1 .
(3.6)
Then, it can be seen that, after a reset jump x(t ) → AR x(t ), the LK functional V = V1 + V2 + V3 (3.1) only changes in its first term V1 , because the integrals in V2 and V3 are not affected. Hence, the condition (ii) 1V = 1V1 ≤ 0 for x ∈ M = Imag T −1 K amounts to the negative semidefiniteness, P¯ ≤ 0, of P¯ = K > T −> (AR PAR − P ) T −1 K ,
There are several Lyapunov–Krasovskii (LK) functionals that can be used for delay-dependent stability conditions. The complete LK functional (Theorem 5.19 in Gu, Kharitonov, & Chen, 2003) has the advantage that it gives a sufficient but also necessary condition for stability; however, it is a distributed functional and requires discretization for practical computation. Another useful result is given by Proposition 5.17 in Gu et al. (2003), with good LMI computability, and it is not too conservative, in general. Finally, the LK functional in Park (1999) has the advantage (detailed in Zhang, Knospe, & Tsiotras, 2001) that the stability conditions can be understood in terms of scaled small-gain, for an uncertain plant. This LK functional is chosen here and takes the form given by Park (1999):
R0 R0
Proof. From Proposition 3.1, what has to be proved is that (i) dV /dt < 0 and (ii) 1V ≤ 0 for x ∈ M. It can be seen that (3.4) implies (i); this has been already proved in Park (1999). So, it only remains to prove that (3.3) implies (ii). Actually, it will be shown that (3.3) and (ii) are equivalent; that is, (3.3) iff (ii). This equivalence will be made explicit by expressing 1V ≤ 0 for x ∈ M as a quadratic form. First, observe from (2.11) that M = Ker C is equivalent to Ker C = Ker T −1 C˜ ; that is,
|
Proof. The detailed proof is omitted for brevity, but it is quite similar to the proof of Theorem 1 in Beker et al. (2004), and to the proof of Proposition 2.1 in Baños and Barreiro (2009). The basic idea is that, from the decreasing behavior, V (xt ) → c ≥ 0 as t → ∞. But if we suppose that c > 0, then xt would always lie outside a ball around the origin, so that necessarily V˙ < −γ < 0, by continuity. But hence V would become negative, and so by contradiction c = 0.
one is V2 (xt ) =
M = Ker T −1 C˜ = Imag T −1 diag (K1 , I , I ) ,
Then, (2.5) is globally asymptotically stable.
R0
> where N = 21 (A + Ad )> P + P (A + Ad ) + WAd + A> d W + Q , and each block ‘ ? ’ is the transpose of the corresponding symmetric block.
1 2
P
< 0,
{z
|
(3.7)
}
Π (P )
which defines Π (P ). However, it can be seen that P¯ = K > Π T −> PT −1
{z
|
P˜
K = K > Π P˜ K .
(3.8)
}
This is true because AR (2.6) and T −1 (2.11) commute, from their block-diagonal structure. Now, partitioning P˜ = (P˜ ij ) for i, j = 1, 2, 3, it holds that
0 Π P˜ = − 0 > P˜ 13
0 0 > P˜ 23
P˜ 13 P˜ 23 . P˜ 33
(3.9)
Introducing K in (3.5):
0 P¯ = K > Π P˜ K = − 0 > P˜ 13 K1
0 0 > P˜ 23
K1> P˜ 13 P˜ 23 . P˜ 33
(3.10)
Recall that (ii) 1V ≤ 0 for x ∈ M is true iff P¯ ≤ 0. And this is equivalent, imposing that the off-diagonal terms are zero, to
− P˜33 ≤ 0,
P˜ 23 = 0,
> and ∃Mβ : P˜ 13 = Mβ C˜ 1 ,
(3.11)
as Imag K1 = Ker C˜ 1 , for some matrix Mβ of appropriate size. Then,
˜ = P, we can say that using T > PT (0, 0, I )P =
> ˜> P˜ 13 , P23 T12 , P˜33
= (Mβ C˜ 1 , 0)T12 , P˜33
(3.12)
and using (2.11) and P˜ 33 = P33 , one immediately obtains (3.12)= (3.3), proving the fact that (3.3) holds iff P¯ ≤ 0 iff 1V ≤ 0 for x ∈ M, which concludes the proof. Remark 1. Notice that, when C2 = 0, then one can take C = C˜ , ˜ which is the case in Beker et al. (2004). Thus, T = In and P = P, the result here extends the previous results not only to the delayed case, but also to a more general form of the reset hypersurface M, with C2 6= 0.
(3.4) Remark 2 (Delay Margin). If the delay h is not given, it is important to estimate the set of values h ∈ Ireset for which (3.2)–(3.4) are
A. Barreiro, A. Baños / Automatica 46 (2010) 216–221
219
feasible. Notice that stability of the reset system is more restrictive than stability of the linear system, so that Ireset ⊂ Ilin . Determining those h ∈ Ilin for which x˙ = Ax + Ad x(t − h) is stable is well known. It is known that h can have an effect of conditional stability: Ilin = [0, η1 ) ∪ (η2 , η3 ) ∪ . . . (ηm−1 , ηm ), with the ηk easily computable (Gu et al., 2003). With this is mind, as Ireset is bounded by Ilin , one can always define a fine mesh of values {hk }Nk=1 covering this set and checking feasibility of the N related LMIs. Furthermore, (3.4) can be written as F + hG < 0, and one can always force negative definiteness by checking Fk + hk Gk < −αk I, for some αk > 0. Then, it is easy to show that we have a stable range (hk −δk , hk +δk ) =: Ik provided that δk = αk /kGk k. Thus, the union ∪k Ik =: Inum provides a numerically guaranteed stable range Inum ⊂ Ireset that can be made arbitrarily close to Ireset by increasing the number of tested LMIs.
Now, it is useful to show that the overall IDDE (2.5) is passive (and stable) by proving that it is the negative feedback connection between BLSio and a certain suitable subsystem R that performs the resetting actions. To show this, consider the ideal resetting system R : y(t ) → −u(t ) =: v(t ) with input y(t ) = Hx(t ) and output −u(t ) = v(t ) given by
3.2. Frequencial conditions based on passivity
Proof. During the continuous mode, both configurations share the same autonomous dynamics. At the reset times tk , the IDDE system resets to zero the component x3 of the controller state x> 23 = > (x> , x ) (the last n states). At the same instants t , the system R 3 k 2 3 behaves as follows. Notice that y(tk ) = Hx(tk ) and so
This section presents frequency conditions related to (3.3) (3.4) to verify stability. The approach is based on the notion of passivity of a system with respect to a certain storage function (the Lyapunov functional). The IDDE system will be interpreted as equivalent to the base-linear subsystem (BLS) in feedback connection with the impulsive resetting subsystem. First, note that condition (3.3) can be rewritten as B> P = H or H > = P B,
(3.13)
with B> = (O, O, I ) , H = Mβ C1 , Mβ C2 , P33 .
x˙ (t ) = A x(t ) + Ad x(t − h) + B u(t ), y(t ) = H x(t ).
(3.17)
(3.18)
where the subindex in the V˙ denotes the system where the time derivative of V is computed. The last expression is easy to deduce and shows that the only influence of u in BLSio affecting V˙ = V˙ 1 +V˙ 2 +V˙ 3 is through the term V˙ 1 . The derivatives of the remaining terms V2 , V3 can be written (Park, 1999) as V˙ 2 = x> (t )Qx(t ) − R0 x> (t − h)Qx(t − h), and V˙ 3 = hF (t ) − −h F (t + θ ) dθ , with F (t ) = G(t )> A> d XAd G(t ) and G(t ) = Ax(t ) + Ad x(t − h). And hence V˙ 2 , V˙ 3 depend only on past values x(t +θ ), θ ≤ 0 of the state and not on the input u(t ), so that (3.18) is true. Then, the passivity condition (3.17) amounts to y> (t )u(t ) = x> (t )H > u(t ) > V˙ BLS (xt ) + x> (t )PBu(t ).
and, by the reset condition Cx = C1 x1 + C2 x2 = 0, y(tk ) = P33 x3 (tk ),
(3.22)
(3.16)
for all u(t ), y(t ), xt solutions to the system. By rebuilding the time derivative of the Lyapunov–Krasovskii functional, it can be seen that V˙ BLSio = V˙ BLS
(3.21)
This is the precise input u(t ) that, applied to the BLS with input matrix B, performs the zero reset of x3 , the last n3 components of the state.
Proof. The sketched arguments are as follows. Recall that the system BLSio is passive with respect to V (xt ), by definition, when
1 1 + x> (t )PB u(t ) + u> (t )B> Px(t ), 2 2
y(tk ) = Mβ C1 x1 (tk ) + Mβ C2 x2 (tk ) + P33 x3 (tk ),
−1 u(t ) ≈ −δ(t − tk )P33 P33 x3 (tk ) = −δ(t − tk )x3 (tk ).
Proposition 3.3. If the conditions (3.3)–(3.4) hold, then the baselinear system BLS io is passive with respect to the storage functional V (xt ) (3.1).
xt 6= 0,
Proposition 3.4. The IDDE system (2.5) is equivalent to {BLSio , −R}, the negative feedback interconnection between the base linear system with inputs and outputs BLS io and the ideal resetting subsystem R.
(3.15)
y> (t ) u(t ) > V˙ BLSio (xt ) ,
where δ(t − tk ) is a Dirac impulse applied at t = tk , the instants when reset conditions hold.
so from (3.20), u(t ) = −v(t ) contains, locally around t = tk , a Dirac delta with weight
and the base linear system with inputs and outputs BLSio :
(3.20)
(3.14)
After this, consider the base linear system BLS : x˙ (t ) = A x(t ) + Ad x(t − h),
−1 R : v(t ) = Σk δ (t − tk ) P33 y(t ),
(3.19)
Now, the only possibility for the previous relation to hold is (setting u = 0) that V˙ BLS (xt ) < 0 (ensured by (3.4)) and (as u(t ) is completely free) that (3.13) is satisfied (equivalent to (3.3)). Notice that (3.4) is only a sufficient condition for V˙ BLS < 0, according to Park (1999). This is the reason why the ‘only if’ direction cannot be proved here.
(3.23)
Now the question is: Could the stability of the IDDE system be proved with passivity techniques? From Proposition 3.2 (internal stability) one can always derive an input–output (I/O) frequency domain result, using the KYP lemma, as in Baños and Barreiro (2009). However, it would be nice to find an independent approach based only on passivity. The main problem on the I/O stability of the loop {BLSio , −R} is that the Dirac impulses are not signals in L2 . Then, the only possibility is to consider L2 approximations of ideal impulses. So, consider the system R : y(t ) → −u(t ) =: v(t ) with input y(t ) = Hx(t ) and output −u(t ) = v(t ) given by −1 R : v(t ) = Σk δ (t − tk ) P33 |y(tk )| ◦ sign (y(t )) ,
(3.24)
where δ (t − tk ) is a rectangular pulse starting at t = tk , with width and height 1/ , and ‘◦’ stands for componentwise vector product. Thus R acts producing pulse trains with height proportional to the value |y(tk )| of its inputs at the reset instants. The function ‘sign’ is included for passivity, because y(t ) may vary fast enough in the interval [tk , tk + ), changing its sign. Connecting R to BLSio , from (3.14)–(3.16), and for t ∈ [tk , tk + ), y˙ (t ) = HAx(t ) + HAd x(t − h) −
|
{z
=: y˙ X (t )
}
|y(tk )| ◦ sign (y(t )) .
(3.25)
To better see how it works, consider the scalar case, with y(tk ) =: yk > 0; then y˙ (t ) = y˙ X (t ) −
yk
sign (y(t )) . (3.26) So, if min (yk ) / max (|˙yX |), then the second term in (3.26) dominates the first one, which is the objective. We make two assumptions: (i) the state x(t ) is restricted to evolve in {x : kx(t )k < Xmax }, and (ii) the resetting is applied only when |y(tk )| > ymin . The bound Xmax can be taken arbitrarily large, covering the
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A. Barreiro, A. Baños / Automatica 46 (2010) 216–221
physically meaningful values of the states, and ymin can be taken arbitrarily small. Both assumptions are necessary, otherwise the dominance of yk / and the resetting effect will be lost. In this way R approaches the ideal reset R (3.20) when → 0, and the resetting action in Proposition 3.4 is correctly applied, because from (3.26) when → 0, the value yk = P33 x3 (tk ) decreases linearly to y(tk + ) = 0 in an arbitrarily fast way. Now, the system R given by (3.4) maps L2e to L2e . However, R is not bounded in L2 since it is possible to find signals in L2 such that its image under R is not in L2 . A similar problem has been discussed in Chen and Francis (1991) in the framework of sampleddata systems. The solution adopted in Chen and Francis (1991) to overcome the problem will be also used here. Basically, the system R is restricted to operate in the space FL2 of filtered L2 signals, F being an LTI system with a strictly proper transfer function and arbitrarily large bandwidth. More precisely, R is restricted to operate on FL?2 = {y ∈ FL2 : |˙y| ≤ M }, where the bound M on y˙ can always be imposed, for any fixed , from the discussion after (3.26). Thus, the system R : FL?2e ⊂ L2e → L2e is bounded; that is, R FL?2 ⊂ L2 . This is a necessary condition for R to be output strictly passive. Proposition 3.5. The system R : FL?2e ⊂ L2e → L2e is output strictly passive. Proof. Recall that a system R : y → v , satisfying T
Z
y> (t )v(t )dt ≥ β + δi 0
T
Z
y> (t )y(t )dt + δo
T
Z
0
v > (t )v(t )dt 0
(3.27) for all y, v, T and some β ≤ 0, is passive when δi , δo ≥ 0, input strictly passive (ISP) when δi > 0, output strictly passive (OSP) when δo > 0, and strictly passive (SP) when δi , δo > 0. Let us suppose here, for simplicity, the scalar case n3 = 1 and P33 = 1. The multivariable case requires a more careful treatment and will be reported elsewhere. First, the L2 norm of the output is
kR yk2 =
Z
(Σk δ (t − tk )|y(tk )|sign (y(t )))2 dt ,
and after some manipulation kR yk2 = (1/) Σk |y(tk )|2 . Then, the inner product hv, yi = hR y, yi is
hR y, yi =
Z
(Σk δ (t − tk )|y(tk )|sign (y(t ))) y(t )dt , Rt
+
so that hR y, yi = (1/)Σk |y(tk )| t k |y(t )| dt. Now, since y is k a filtered L2 signal, that is y ∈ FL?2e , |˙y| is bounded; then one can
Rt
+
obtain t k |y(t )|dt ≥ α() |y(tk )|, where the coefficient α() > k 0 can be obtained from (3.26), R by taking the worst-case evolution of y(t ) (least integral value |y|), as a function of the corresponding bounds |yk | ≥ ymin and |˙yX | ≤ (kHAk + kHAd k) Xmax . It also becomes clear that when → 0 the evolution for (t , yR(t )) in (3.26) is a straight line from (tk , yk ) to (tk + , 0), so that |y| = yk /2, meaning that α() → 1/2 when is small enough. Finally, comparing hR y, yi and kR yk2 ,
hR y, yi ≥ α Σk |y(tk )|2 = αkR yk2 ≥ δkR yk2 , and thus, from (3.27), R is output strictly passive for some δ such that 0 < δ ≤ α . Remark 3. An important consequence of the last result is that the stability condition for the feedback system {BLSio , −R } can be translated into a frequencial condition on BLSio . To see this, recall that a negative feedback system {G1 , −G2 } is finite-gain stable if any of (i)–(iii) holds (Carrasco, Baños, & van der Schaft, 2008): (i) G1 , G2 are ISP, (ii) G1 , G2 are OSP, (iii) G1 is SP and G2 is passive.
Clearly, we are interested in case (ii), so if BLSio is OSP, then closed loop stability (finite gain) is guaranteed. Recall also that (Carrasco et al., 2008) a stable and controllable LTI system G(s) is OSP if there exists some δo > 0 such that G(jω) + G? (jω) ≥ δo kG(jω)k2 . In the scalar case, this means that G(jω) lies inside a circle with center s = 1/(2δo ) and radius r = 1/(2δo ). In summary, the required stability condition is Hβ (jω) + Hβ? (jω) ≥
δkHβ (jω)k2 , δ > 0; that is, an OSP condition on −1 B. Hβ (jω) = H jωI − A − Ad e−jωh
(3.28)
In this way, this is a generalization of the results in Beker et al. (2004). In the particular case whenAd → 0 (without time delay), and when H = Mβ C1 , Mβ C2 , P33 → Mβ C1 , O, P33 , one gets the so-called Hβ SPR (positive realness) condition in Beker et al. (2004). Strictly speaking, this passivity analysis ensures stability of the loop {Hβ (s), −R } in the I/O sense. In general, recovering internal stability would require mild assumptions (controllability, detectability, . . . ) However, the passivity result here is presented as an extra interpretation of an original LK analysis which directly ensures internal asymptotic stability. Remark 4 (Robustness in Terms of Small-Gain and Positive Uncertainties). One nice feature of the LK functional (3.1) (Park, 1999) is that it provides an interpretation within a scaled small-gain framework (Zhang et al., 2001). Similarly, the previous passivity analysis can be recast in a robust stability setting. These two results could be combined, providing explicitly which form of uncertainty can be tolerated. It is not included, for brevity, but it is another advantage of the proposed approach. Remark 5 (Comparison with the KYP Approach). One standard tool for passing from time-domain LMIs to frequency-domain conditions is the Kalman–Yacubovitch–Popov (KYP) lemma, used in Baños and Barreiro (2009). The KYP lemma here gives rise to big matrix conditions, and is a more involved approach. 4. Example Consider a feedback system according to the setup of Fig. 1. Here the plant has a transfer function e−hs P0 (s) given by (the example is adapted from Beker et al. (2004)) s+1 , (4.1) s(s + 0.2) and the feedback compensator K is a FORE compensator with base LTI system given by Kbl (s) = 1/(s + 1). It can be shown that the base LTI control system is stable for delays h < 0.2. In the following, the passivity result in Section 3 is used to show that the reset action does not destabilize the system, by checking that (3.28) is OSP. A delay h = 0.15 is chosen. In this simple case the plant has two states and the reset compensator has one state only; thus n1 = 2, n2 = 0, and n3 = 1. Let us use the realizations of P0 (s) and Kbl (s): e−hs P0 (s) = e−hs
Ap =
−0.2 1
0 , 0
Bp =
1 , 0
Cp = 1
1
(4.2)
and Ac = −1, Bc = 1, Cc = 1. Here Mβ = (β) and P33 are scalar, and P33 can be chosen as P33 = 1 without loss of generality. As a result, matrices H and B are given by (3.14): H = Mβ Cp
P33 = β
β
1 ,
B> = 0
0
1 .
(4.3)
With the above values, system (3.28) is OSP for values of β between 0.29 and 0.56. As a result, the stability of the reset control system is guaranteed, in this case for h = 0.15. An identical procedure shows that the reset action never destabilizes a stable base LTI control system and its stability limit is h = 0.2, the same limit as the base LTI control system. Fig. 2 plots the time response, showing the typical fast transient and not excessive overshoot gained with reset.
A. Barreiro, A. Baños / Automatica 46 (2010) 216–221
Fig. 2. Time response yp (t ) (top) and yc (t ) (bottom).
5. Conclusions The stability analysis developed in this work is based on the Lyapunov functional V (xt ) (3.1), for which it is required (Proposition 3.1) that (i) V˙ (xt ) < 0 and that (ii) 1V (xt ) ≤ 0 at the reset surface. For the first condition, (i) V˙ (xt ) < 0, to avoid conservativeness, we have used a quite general functional V (xt ) (3.1) containing several free matrix parameters, with good accuracy and low computational cost. For the second condition, (ii) 1V (xt ) ≤ 0, we have contributed with the passivity analysis. The main idea is the interpretation of the overall system as the negative feedback of the base system BLSio with the impulsive resetting subsystem R. This interpretation puts the problem in a well-known framework and facilitates further treatment using frequency domain results. The case when the base system is unstable and resetting is used to stabilize it is not treated in this paper, and it cannot be treated using Proposition 3.1. It requires allowing V˙ (xt ) or 1V (xt ) to be positive sometimes, provided that, on the average, V decreases. A good idea is to bound the time intervals between resetting instants and use a discrete-time analysis. This strategy was applied in Baños et al. (2007) to unstable plants without delay. The extension to delayed cases is an interesting open problem. Finally, Carrasco et al. (2008) deals with the stability of nonlinear reset systems, using passivity and dissipativity techniques. One of the application areas where time delays are more critical is the field of teleoperation. In Fernández, Barreiro, Baños, and Carrasco (2008), the reset strategy is applied to simulated teleoperation systems. Achieving an actual performance improvement in real teleoperated systems in one of the most promising advantages of reset control of time-delay systems. References Åström, K. J. (2000). Limitations on control system performance. European Journal of Control, 6, 2–20. Bainov, D. D., & Simeonov, (1989). Systems with impulse effect: Stability, theory and applications. Chichester: Ellis Horwood. Baños, A., & Barreiro, A. (2009). Delay-independent stability of reset systems. IEEE Transactions on Automatic Control, 341–346. Baños, A., Carrasco, J., & Barreiro, A. (2007). Reset-times dependent stability of reset control with unstable base systems. In Proceedings of the IEEE Int. Symp. on Industrial Electronics (ISIE), Vigo, Spain, June 4-7. Beker, O, Hollot, C. V., Chait, Y., & Han, H. (2004). Fundamental properties of reset control systems. Automatica, 40, 905–915.
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Carrasco, J., Baños, A., & van der Schaft, A. (2008). A passivity approach to reset control on nonlinear systems, IEEE Ind. Electr. Conference (IECON), Orlando, Florida, 10-13 November. Chen, T., & Francis, B. A. (1991). Input–output stability of sampled data systems. IEEE Transactions on Automatic Control, 36(1). Clegg, J. C. (1958). A nonlinear integrator for servomechanism. Transactions A.I.E.E.m, Part II, 77, 41–42. De la Sen, M., & Luo, N. (2003). A note on the stability of linear time-delay systems with impulsive inputs. IEEE Transactions on Circuits and Systems, 50(1), 149–152. Fernández, A., Barreiro, A., Baños, A., & Carrasco, J. (2008). Reset control for passive teleoperation. IEEE Ind. Electr. Conf. (IECON), Orlando, Florida, 10-13 November 2008. Hespanha, J. P., Liberzon, D., & Teel, A. R. (2008). Lyapunov conditions for input to state stability of impulsive systems. Automatica, 44, 2735–2744. Horowitz, I. M., & Rosenbaum, (1975). Nonlinear design for cost of feedback reduction in systems with large parameter uncertainty. International Journal of Control, 24(6), 977–1001. Gu, K., Kharitonov, V. L., & Chen, J. (2003). Stability of time-delay systems. Boston: Birkhauser. Guo, Y., Wang, Y., Zheng, J., & Xie, L. (2007). Stability analysis, design and application of reset control systems. In Proceedings of the IEEE Int. Conf. on Control and Automation, Guangzhou, China, May 30–June 1, 2007. Haddad, W. M., Nersesov, S. G., & Chellaboina, V. S. (2003). Energy-based control for hybrid port-controlled Hamiltonian Systems. Automatica, 39, 1425–1435. Lakshmikantham, V., Bainov, D. D., & Simeonov, P. S. (1989). Theory of impulsive differential equations. Singapore: World Scientific. Liu, X., & Ballinger, G (2001). Uniform asymptotic stability of impulsive delay differential equations. Computers and Mathematics with applications, 41, 903–915. Liu, X., Shen, X., Zhang, Y., & Wang, Q. (2007). Stability criteria for impulsive systems with time delay and unstable system matrices. IEEE Transactions on Circuits and Sytems, 54(10). Liu, X., & Wang, Q. (2006). Stability of nontrivial solution of delay differential equations with state-dependent impulses. Applied Mathematics and Computation, 174, 271–288. Krishman, K. R, & Horowitz, I. M. (1974). Synthesis of a nonlinear feedback system with significant plant-ignorance for prescribed system tolerances. International Journal of Control, 19(4), 689–706. Nešić, D., Zaccarian, L., & Teel, A. R. (2005). Stability properties of reset systems. Automatica, 44, 2019–2026. Park, P. (1999). A delay-dependent stability criterion for systems with uncertain time-invariant delays. IEEE Transactions on Automatic Control, 44, 876–877. Yang, T. (2001). Lecture Notes in Control and Information Sciences: Vol. 272. Impulsive Control Theory. Berlin: Springer. Zhang, J., Knospe, C. R., & Tsiotras, P. (2001). Stability of time-delay systems: Equivalence between Lyapunov and scaled small-gain conditions’. IEEE Transactions on Automatic Control, 46, 482–486.
Antonio Barreiro was born in Vigo, Spain in 1959. He received his degrees of Ingeniero and Doctor Ingeniero Industrial from the Polytechnic University of Madrid (UPM) in 1984 and 1989, respectively. From 1984 to 1987 he was with the Departamento de Matemática Aplicada of the E.T.S. Ingenieros Industriales of the UPM. Since 1987 he has been with the Departamento de Ingeniería de Sistemas y Automática of the University of Vigo, where he is now Professor in the area of Automatic Control. His research interests include nonlinear and robust stability, time-delay systems and reset control, with applications in networked control and teleoperation systems.
Alfonso Baños was born in Córdoba, Spain in 1965. He received his degrees of Licenciado and Doctor in Physics from the University of Madrid (Complutense) in 1987 and 1991, respectively. From 1988 to 1992 he was with the Instituto de Automática Industrial (C.S.I.C.) in Madrid, where he pursued research in nonlinear control and robotics. In 1992 he joined the University of Murcia, where he is currently Professor in the area of Automatic Control. He has also held visiting appointments at the University of Strathclyde, the University of Minnesota at Minneapolis, and the University of California at Berkeley. His research interests include robust and nonlinear control, and reset control, with applications in process control and networked control systems.
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Brief paper
Delay-dependent stability of reset systemsI Antonio Barreiro a,∗ , Alfonso Baños b a
E.T.S. Ingenieros Industriales, University of Vigo, 36200 Vigo, Spain
b
Facultad de Informática, University of Murcia, 30071 Murcia, Spain
article
info
Article history: Received 9 July 2008 Received in revised form 23 July 2009 Accepted 21 October 2009 Available online 18 November 2009 Keywords: Reset control Hybrid systems Time-delay systems Passivity Lyapunov–Krasovskii functionals
abstract This work presents results on the stability of time-delay systems under reset control. The case of delaydependent stability is addressed, by developing a generalization of previous stability results for reset systems without delay, and also a generalization of the delay-independent case. The stability results are derived by using appropriate Lyapunov–Krasovskii functionals, obtaining LMI (Linear Matrix Inequality) conditions and showing connections with passivity and positive realness. The stability conditions guarantee that the reset action does not destabilize the base LTI (Linear Time Invariant) system. Several interpretations are given for these conditions in terms of impulsive control, which provide insights into the potentials of reset control. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The idea of reset control and in particular Clegg’s integrator (Clegg, 1958) and the first-order reset element (FORE) (Horowitz & Rosenbaum, 1975; Krishman & Horowitz, 1974) are attempts to achieve fast and robust control solutions for problems under linear limitations, which are particularly severe for plants with right-half plane poles or zeros, or with time delays (Åström, 2000). The implementation of reset control is very simple: it consists of resetting the state (or part of it) of a feedback LTI compensator (the ‘base’ LTI compensator) at every instant in which the tracking error is zero. A design approach (Horowitz & Rosenbaum, 1975; Krishman & Horowitz, 1974) is to tune the base LTI controller so that the base loop becomes stable, with fast transient and large bandwidth. Then, a reset action is included to achieve reduced overshoot and large stability margins. However, this method should be carefully applied since the reset may destabilize a well-designed LTI system. A solution to this problem has been reported in Beker, Hollot, Chait, and Han (2004) for finite-dimensional LTI plants. The result is a frequency domain condition, the Hβ condition, that guarantees closed loop stability of the reset system. Since then, a number of
I Work supported by MEC, Spain, under project DPI-2007 66455. The material in this paper was partially presented at the American Control Conference, New York, July 11–12, 2007. This paper was recommended for publication in revised form by Associate Editor Zhihua Qu under the direction of Editor Andrew R. Teel. ∗ Corresponding author. Tel.: +34 986 812232; fax: +34 986 814014. E-mail addresses: [email protected] (A. Barreiro), [email protected] (A. Baños).
0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2009.10.029
useful results have appeared for reset systems (Baños, Carrasco, & Barreiro, 2007; Hespanha, Liberzon, & Teel, 2008; Guo, Wang, Zheng, & Xie, 2007; Haddad, Nersesov, & Chellaboina, 2003; Nešić, Zaccarian, & Teel, 2005). However, none of them considered timedelay plants. As reset control is able to overcome fundamental limitations, and time delay is one source of such limitations, it is then of great interest to study the problem of delayed reset systems. It should be emphasized that this is extremely difficult: the combination of reset and delay moves the problem to the field of hybrid and distributed systems. Even the issue of existence and uniqueness of solutions may cause trouble unless special care is taken, for example, by means of the time-regularization procedure (Nešić et al., 2005). It should be noted that stability of impulsive systems is a very active research area. Two classical monographs are Bainov and Simeonov (1989) and Lakshmikantham, Bainov, and Simeonov (1989). In general, as reported in Bainov and Simeonov (1989), impulsive systems may be classified in (i) systems with impulses at fixed instants, (ii) systems with impulse effect at variable instants, an (iii) autonomous systems with impulse effects. Both (ii) and (iii) have state-dependent impulses, but there is an important difference: the reset surface is time independent in case (iii). Traditionally, most of the research effort has been dedicated to cases (i) and (ii), and thus case (iii), which includes reset control systems, is less developed. In addition, in the control field the main effort has been done for systems without time delays (Haddad et al., 2003; Yang, 2001). For the case of impulsive delay systems, available results in the literature are concentrated on case (i) (see De la Sen and Luo (2003), Liu and Ballinger (2001), Liu, Shen, Zhang,
A. Barreiro, A. Baños / Automatica 46 (2010) 216–221
and Wang (2007), and references therein), and case (ii) (see Liu and Wang (2006)). The objective of the present paper is to analyze the stability of time-delay systems under reset control. In particular, we try to find generalizations of the results in Beker et al. (2004) for reset systems without time delay, and also generalizations of the results in Baños and Barreiro (2009) for the delay-independent case. The generalization of these results suggests, for continuity, using the same common basic stability tool (Proposition 3.1 below), a theorem which requires that the energy (Lyapunov) must always decrease during the continuous and the impulsive behaviors. This basic result may be conservative, in general, and within the area of impulsive systems there is a vast amount of literature offering alternative techniques. At the same time, notice that many reset design approaches (Beker et al., 2004; Horowitz & Rosenbaum, 1975; Haddad et al., 2003; Krishman & Horowitz, 1974) are based on the useful idea of tuning the base system for a fast transient, underdamped but stable, and introducing reset for adding extra damping and decreasing overshoot. Within this idea, both the continuous and the impulsive mode must be stable, and so the basic tool (Proposition 3.1) is very well suited and adequate. Once the basic criterion (Proposition 3.1) is chosen, the motivation here is to explore its limits, in particular by obtaining generalizations of Baños and Barreiro (2009) and Beker et al. (2004). Following this, the original contributions of this paper are as follows. Firstly, the so-called Hβ condition in Beker et al. (2004) is extended to time-delay systems, as in Baños and Barreiro (2009), but now obtaining delay-dependent results, and with a more general structure C = (C1 , C2 , 0) of the output matrix (Proposition 3.2). Secondly, for passing from the time domain (Lyapunov and LMIs) to the frequency domain, an original idea is presented, based on interpreting resetting as an impulsive control around a certain base linear system (Propositions 3.3 and 3.4). This idea is different from the technique used in Baños and Barreiro (2009), based on the Kalman–Yacubovitch–Popov (KYP) lemma. Thirdly, the application of the impulse approach requires a careful treatment of the associated passivity conditions, as Dirac impulses are not L2 signals. This is analyzed in Proposition 3.5 and Remark 3. The organization of the paper is as follows. After introducing the problem in Section 2, the main stability results are given in Section 3. First, in 3.1, a Lyapunov–Krasovskii functional is used to derive delay-dependent stability conditions in the LMI form. Then, in 3.2, a second approach is developed based on passivity analysis. Section 4 presents an example and Section 5 gives our final conclusions. 2. Problem motivation and statement Consider a control system formed by a plant P0 and a reset controller K (both single-input–single-output):
P0 :
x˙ p = Ap xp + Bp up yp = Cp xp ,
(2.1)
K :
x˙ c = Ac xc + Bc uc xc ( t + ) = A ρ xc ( t ) y = C x + D u , c c c c c
(2.2)
where the second equation of the controller represents the impulsive or reset actions, applied when a certain reset condition, specified later, holds. If the connection from P0 to K (from K to P0 ) is affected by a delay h1 (h2 ), then it is easy to see that the total delay, h = h1 + h2 , can be moved to the plant input or output. For instance, move the delay to the plant input, as shown in Fig. 1. As the main interest is stability, let us suppose that the exogenous loop signals are zero (r = d = n = 0), so that the closed loop connections are up (t ) = yc (t − h),
uc (t ) = − yp (t ),
(2.3)
217
Fig. 1. Reset control system setup.
and the closed loop system is easily described by
x˙ p (t ) x˙ c (t )
x (t ) =A p xc (t )
x p ( t − h) , x c ( t − h)
+ Ad
(2.4)
and by impulsive dynamics xc (t + ) = Aρ xc (t ). With this in mind, we generalize to the system given by the impulsive-delayed differential equation x˙ (t ) = A x(t ) + Ad x(t − h), x(t + ) = AR x(t ),
IDDE
iff x(t ) 6∈ M , iff x(t ) ∈ M ,
(2.5)
for arbitrary A and Ad . It remains to introduce AR and M. Let us suppose that only the second part of the controller states are reset to zero; that is, Aρ = diag(1, . . . , 1, 0, . . . , 0). Then AR takes the form Beker et al. (2004) AR = diag In12 , 0n3 ,
(2.6)
so, the reset action acts over the last n3 states of the state x ∈ Rn . These dimensions are defined by n = n1 + n2 + n3 =
| {z } n23
n + n +n , | 1 {z }2 3
(2.7)
n12
where n1 (n23 ) is the number of states of the plant (controller). Accordingly, partition the state vector: > > x> = ( x> 1 , x2 , x3 ).
(2.8)
|{z} | {z } x> p
x> c
In principle, it is considered that the reset is applied at the time t when x(t ) hits the hypersurface M = x ∈ Rn : Cx = 0 ,
(2.9)
where the linear resetting condition has the structure 0 = Cx(t ) = C1 x1 (t ) + C2 x2 (t ).
(2.10)
As it is desired that 0 = Cx defines a hypersurface, then C is a row, C ∈ R1,n , and we assume here for C the structure C = (C1 , C2 , 0), which is more general than C = (C˜ 1 , 0, 0), the one adopted in Beker et al. (2004). For comparison with this last case, notice that there exists an invertible transformation T ∈ Rn,n such that C˜
T
C }| z }| { z }| { z { (C1 , C2 , 0) = (C˜ 1 , 0, 0) diag T12 , In3 .
(2.11)
One possible solution is C˜ 1 = (1, 0, . . . , 0) and T12 with its first row equal to (C1 , C2 ) and the remaining rows completed for full rank. In this way, (2.5), (2.6) and (2.9) define the system that we are going to study. It could be possible to adopt, following Nešić et al. (2005), the useful idea of time regularization, as a strategy for ensuring well-posedness and avoiding beating and Zenoness. However, this is not addressed here, for the sake of brevity and because the stability problem is essentially the same. 3. Stability analysis In this section, the stability of the delayed reset control system (2.5) is analyzed. Note that it is important to distinguish between the (lumped) state x(t ) ∈ Rn and the true (distributed) state xt : [−h, 0] → Rn , where xt (θ ) = x(t + θ ), θ ∈ [−h, 0]. In what follows, C ([−h, 0], Rn ) stands for the set of piecewise
218
A. Barreiro, A. Baños / Automatica 46 (2010) 216–221
continuous functions xt (·) mapping [−h, 0] to Rn , with a finite number of discontinuity points. To address stability, consider certain Lyapunov–Krasovskii candidate functionals V (xt ), and use as a basic tool the following proposition (Beker et al., 2004). Proposition 3.1. Let V (xt ) : C ([−h, 0], Rn ) → R be continuously differentiable, positive-definite, radially unbounded, such that, along the solutions of (2.5), d dt
V (xt ) < 0,
xt 6= 0,
1 V (xt ) = V (xt + ) − V (xt ) ≤ 0,
x(t ) ∈ M .
3.1. Stability analysis based on LMIs
V (xt ) = V1 (xt ) + V2 (xt ) + V3 (xt ),
(3.1)
where the first term is V1 (xt ) =
>
−h >
1 2
x (t ) P x(t ), the second >
x (t + θ ) Q x(t + θ ) dθ , and V3 (xt ) = >
Ad XAd G xt +ξ dξ dθ , with G xt +ξ = Ax(t + −h θ G xt +ξ ξ ) + Ad x(t + ξ − h). Thus, by applying Proposition 3.1 to V (xt ),
we obtain sufficient conditions for global asymptotical stability. The requirement that d [V (xt )] /dt < 0 has been already treated in Park (1999), and it is finally expressed in LMI form. The requirement that 1V (xt ) ≤ 0 when x(t ) ∈ M can be addressed in a special form, related to the one in Theorem 5 in Beker et al. (2004). By combining these two approaches, we obtain the following proposition. Proposition 3.2. The system (2.5) is globally and asymptotically stable, if there exist Q > 0, V > 0, W , and P = P > = Pij i,j=1,2,3 > 0,
(3.2)
with
(0, 0, I ) P = Mβ (C1 , C2 ), P33 ,
(3.3)
for some matrix Mβ of adequate size, and such that
N
? ? ?
> >
−WAd
A Ad V
−Q ?
> >
0
Ad Ad V −V 0
h W+ 0 0 −V
{z
(3.5)
}
K
where I is the n3 × n3 identity and K1 is such that Imag K1 = Ker C˜ 1 .
(3.6)
Then, it can be seen that, after a reset jump x(t ) → AR x(t ), the LK functional V = V1 + V2 + V3 (3.1) only changes in its first term V1 , because the integrals in V2 and V3 are not affected. Hence, the condition (ii) 1V = 1V1 ≤ 0 for x ∈ M = Imag T −1 K amounts to the negative semidefiniteness, P¯ ≤ 0, of P¯ = K > T −> (AR PAR − P ) T −1 K ,
There are several Lyapunov–Krasovskii (LK) functionals that can be used for delay-dependent stability conditions. The complete LK functional (Theorem 5.19 in Gu, Kharitonov, & Chen, 2003) has the advantage that it gives a sufficient but also necessary condition for stability; however, it is a distributed functional and requires discretization for practical computation. Another useful result is given by Proposition 5.17 in Gu et al. (2003), with good LMI computability, and it is not too conservative, in general. Finally, the LK functional in Park (1999) has the advantage (detailed in Zhang, Knospe, & Tsiotras, 2001) that the stability conditions can be understood in terms of scaled small-gain, for an uncertain plant. This LK functional is chosen here and takes the form given by Park (1999):
R0 R0
Proof. From Proposition 3.1, what has to be proved is that (i) dV /dt < 0 and (ii) 1V ≤ 0 for x ∈ M. It can be seen that (3.4) implies (i); this has been already proved in Park (1999). So, it only remains to prove that (3.3) implies (ii). Actually, it will be shown that (3.3) and (ii) are equivalent; that is, (3.3) iff (ii). This equivalence will be made explicit by expressing 1V ≤ 0 for x ∈ M as a quadratic form. First, observe from (2.11) that M = Ker C is equivalent to Ker C = Ker T −1 C˜ ; that is,
|
Proof. The detailed proof is omitted for brevity, but it is quite similar to the proof of Theorem 1 in Beker et al. (2004), and to the proof of Proposition 2.1 in Baños and Barreiro (2009). The basic idea is that, from the decreasing behavior, V (xt ) → c ≥ 0 as t → ∞. But if we suppose that c > 0, then xt would always lie outside a ball around the origin, so that necessarily V˙ < −γ < 0, by continuity. But hence V would become negative, and so by contradiction c = 0.
one is V2 (xt ) =
M = Ker T −1 C˜ = Imag T −1 diag (K1 , I , I ) ,
Then, (2.5) is globally asymptotically stable.
R0
> where N = 21 (A + Ad )> P + P (A + Ad ) + WAd + A> d W + Q , and each block ‘ ? ’ is the transpose of the corresponding symmetric block.
1 2
P
< 0,
{z
|
(3.7)
}
Π (P )
which defines Π (P ). However, it can be seen that P¯ = K > Π T −> PT −1
{z
|
P˜
K = K > Π P˜ K .
(3.8)
}
This is true because AR (2.6) and T −1 (2.11) commute, from their block-diagonal structure. Now, partitioning P˜ = (P˜ ij ) for i, j = 1, 2, 3, it holds that
0 Π P˜ = − 0 > P˜ 13
0 0 > P˜ 23
P˜ 13 P˜ 23 . P˜ 33
(3.9)
Introducing K in (3.5):
0 P¯ = K > Π P˜ K = − 0 > P˜ 13 K1
0 0 > P˜ 23
K1> P˜ 13 P˜ 23 . P˜ 33
(3.10)
Recall that (ii) 1V ≤ 0 for x ∈ M is true iff P¯ ≤ 0. And this is equivalent, imposing that the off-diagonal terms are zero, to
− P˜33 ≤ 0,
P˜ 23 = 0,
> and ∃Mβ : P˜ 13 = Mβ C˜ 1 ,
(3.11)
as Imag K1 = Ker C˜ 1 , for some matrix Mβ of appropriate size. Then,
˜ = P, we can say that using T > PT (0, 0, I )P =
> ˜> P˜ 13 , P23 T12 , P˜33
= (Mβ C˜ 1 , 0)T12 , P˜33
(3.12)
and using (2.11) and P˜ 33 = P33 , one immediately obtains (3.12)= (3.3), proving the fact that (3.3) holds iff P¯ ≤ 0 iff 1V ≤ 0 for x ∈ M, which concludes the proof. Remark 1. Notice that, when C2 = 0, then one can take C = C˜ , ˜ which is the case in Beker et al. (2004). Thus, T = In and P = P, the result here extends the previous results not only to the delayed case, but also to a more general form of the reset hypersurface M, with C2 6= 0.
(3.4) Remark 2 (Delay Margin). If the delay h is not given, it is important to estimate the set of values h ∈ Ireset for which (3.2)–(3.4) are
A. Barreiro, A. Baños / Automatica 46 (2010) 216–221
219
feasible. Notice that stability of the reset system is more restrictive than stability of the linear system, so that Ireset ⊂ Ilin . Determining those h ∈ Ilin for which x˙ = Ax + Ad x(t − h) is stable is well known. It is known that h can have an effect of conditional stability: Ilin = [0, η1 ) ∪ (η2 , η3 ) ∪ . . . (ηm−1 , ηm ), with the ηk easily computable (Gu et al., 2003). With this is mind, as Ireset is bounded by Ilin , one can always define a fine mesh of values {hk }Nk=1 covering this set and checking feasibility of the N related LMIs. Furthermore, (3.4) can be written as F + hG < 0, and one can always force negative definiteness by checking Fk + hk Gk < −αk I, for some αk > 0. Then, it is easy to show that we have a stable range (hk −δk , hk +δk ) =: Ik provided that δk = αk /kGk k. Thus, the union ∪k Ik =: Inum provides a numerically guaranteed stable range Inum ⊂ Ireset that can be made arbitrarily close to Ireset by increasing the number of tested LMIs.
Now, it is useful to show that the overall IDDE (2.5) is passive (and stable) by proving that it is the negative feedback connection between BLSio and a certain suitable subsystem R that performs the resetting actions. To show this, consider the ideal resetting system R : y(t ) → −u(t ) =: v(t ) with input y(t ) = Hx(t ) and output −u(t ) = v(t ) given by
3.2. Frequencial conditions based on passivity
Proof. During the continuous mode, both configurations share the same autonomous dynamics. At the reset times tk , the IDDE system resets to zero the component x3 of the controller state x> 23 = > (x> , x ) (the last n states). At the same instants t , the system R 3 k 2 3 behaves as follows. Notice that y(tk ) = Hx(tk ) and so
This section presents frequency conditions related to (3.3) (3.4) to verify stability. The approach is based on the notion of passivity of a system with respect to a certain storage function (the Lyapunov functional). The IDDE system will be interpreted as equivalent to the base-linear subsystem (BLS) in feedback connection with the impulsive resetting subsystem. First, note that condition (3.3) can be rewritten as B> P = H or H > = P B,
(3.13)
with B> = (O, O, I ) , H = Mβ C1 , Mβ C2 , P33 .
x˙ (t ) = A x(t ) + Ad x(t − h) + B u(t ), y(t ) = H x(t ).
(3.17)
(3.18)
where the subindex in the V˙ denotes the system where the time derivative of V is computed. The last expression is easy to deduce and shows that the only influence of u in BLSio affecting V˙ = V˙ 1 +V˙ 2 +V˙ 3 is through the term V˙ 1 . The derivatives of the remaining terms V2 , V3 can be written (Park, 1999) as V˙ 2 = x> (t )Qx(t ) − R0 x> (t − h)Qx(t − h), and V˙ 3 = hF (t ) − −h F (t + θ ) dθ , with F (t ) = G(t )> A> d XAd G(t ) and G(t ) = Ax(t ) + Ad x(t − h). And hence V˙ 2 , V˙ 3 depend only on past values x(t +θ ), θ ≤ 0 of the state and not on the input u(t ), so that (3.18) is true. Then, the passivity condition (3.17) amounts to y> (t )u(t ) = x> (t )H > u(t ) > V˙ BLS (xt ) + x> (t )PBu(t ).
and, by the reset condition Cx = C1 x1 + C2 x2 = 0, y(tk ) = P33 x3 (tk ),
(3.22)
(3.16)
for all u(t ), y(t ), xt solutions to the system. By rebuilding the time derivative of the Lyapunov–Krasovskii functional, it can be seen that V˙ BLSio = V˙ BLS
(3.21)
This is the precise input u(t ) that, applied to the BLS with input matrix B, performs the zero reset of x3 , the last n3 components of the state.
Proof. The sketched arguments are as follows. Recall that the system BLSio is passive with respect to V (xt ), by definition, when
1 1 + x> (t )PB u(t ) + u> (t )B> Px(t ), 2 2
y(tk ) = Mβ C1 x1 (tk ) + Mβ C2 x2 (tk ) + P33 x3 (tk ),
−1 u(t ) ≈ −δ(t − tk )P33 P33 x3 (tk ) = −δ(t − tk )x3 (tk ).
Proposition 3.3. If the conditions (3.3)–(3.4) hold, then the baselinear system BLS io is passive with respect to the storage functional V (xt ) (3.1).
xt 6= 0,
Proposition 3.4. The IDDE system (2.5) is equivalent to {BLSio , −R}, the negative feedback interconnection between the base linear system with inputs and outputs BLS io and the ideal resetting subsystem R.
(3.15)
y> (t ) u(t ) > V˙ BLSio (xt ) ,
where δ(t − tk ) is a Dirac impulse applied at t = tk , the instants when reset conditions hold.
so from (3.20), u(t ) = −v(t ) contains, locally around t = tk , a Dirac delta with weight
and the base linear system with inputs and outputs BLSio :
(3.20)
(3.14)
After this, consider the base linear system BLS : x˙ (t ) = A x(t ) + Ad x(t − h),
−1 R : v(t ) = Σk δ (t − tk ) P33 y(t ),
(3.19)
Now, the only possibility for the previous relation to hold is (setting u = 0) that V˙ BLS (xt ) < 0 (ensured by (3.4)) and (as u(t ) is completely free) that (3.13) is satisfied (equivalent to (3.3)). Notice that (3.4) is only a sufficient condition for V˙ BLS < 0, according to Park (1999). This is the reason why the ‘only if’ direction cannot be proved here.
(3.23)
Now the question is: Could the stability of the IDDE system be proved with passivity techniques? From Proposition 3.2 (internal stability) one can always derive an input–output (I/O) frequency domain result, using the KYP lemma, as in Baños and Barreiro (2009). However, it would be nice to find an independent approach based only on passivity. The main problem on the I/O stability of the loop {BLSio , −R} is that the Dirac impulses are not signals in L2 . Then, the only possibility is to consider L2 approximations of ideal impulses. So, consider the system R : y(t ) → −u(t ) =: v(t ) with input y(t ) = Hx(t ) and output −u(t ) = v(t ) given by −1 R : v(t ) = Σk δ (t − tk ) P33 |y(tk )| ◦ sign (y(t )) ,
(3.24)
where δ (t − tk ) is a rectangular pulse starting at t = tk , with width and height 1/ , and ‘◦’ stands for componentwise vector product. Thus R acts producing pulse trains with height proportional to the value |y(tk )| of its inputs at the reset instants. The function ‘sign’ is included for passivity, because y(t ) may vary fast enough in the interval [tk , tk + ), changing its sign. Connecting R to BLSio , from (3.14)–(3.16), and for t ∈ [tk , tk + ), y˙ (t ) = HAx(t ) + HAd x(t − h) −
|
{z
=: y˙ X (t )
}
|y(tk )| ◦ sign (y(t )) .
(3.25)
To better see how it works, consider the scalar case, with y(tk ) =: yk > 0; then y˙ (t ) = y˙ X (t ) −
yk
sign (y(t )) . (3.26) So, if min (yk ) / max (|˙yX |), then the second term in (3.26) dominates the first one, which is the objective. We make two assumptions: (i) the state x(t ) is restricted to evolve in {x : kx(t )k < Xmax }, and (ii) the resetting is applied only when |y(tk )| > ymin . The bound Xmax can be taken arbitrarily large, covering the
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physically meaningful values of the states, and ymin can be taken arbitrarily small. Both assumptions are necessary, otherwise the dominance of yk / and the resetting effect will be lost. In this way R approaches the ideal reset R (3.20) when → 0, and the resetting action in Proposition 3.4 is correctly applied, because from (3.26) when → 0, the value yk = P33 x3 (tk ) decreases linearly to y(tk + ) = 0 in an arbitrarily fast way. Now, the system R given by (3.4) maps L2e to L2e . However, R is not bounded in L2 since it is possible to find signals in L2 such that its image under R is not in L2 . A similar problem has been discussed in Chen and Francis (1991) in the framework of sampleddata systems. The solution adopted in Chen and Francis (1991) to overcome the problem will be also used here. Basically, the system R is restricted to operate in the space FL2 of filtered L2 signals, F being an LTI system with a strictly proper transfer function and arbitrarily large bandwidth. More precisely, R is restricted to operate on FL?2 = {y ∈ FL2 : |˙y| ≤ M }, where the bound M on y˙ can always be imposed, for any fixed , from the discussion after (3.26). Thus, the system R : FL?2e ⊂ L2e → L2e is bounded; that is, R FL?2 ⊂ L2 . This is a necessary condition for R to be output strictly passive. Proposition 3.5. The system R : FL?2e ⊂ L2e → L2e is output strictly passive. Proof. Recall that a system R : y → v , satisfying T
Z
y> (t )v(t )dt ≥ β + δi 0
T
Z
y> (t )y(t )dt + δo
T
Z
0
v > (t )v(t )dt 0
(3.27) for all y, v, T and some β ≤ 0, is passive when δi , δo ≥ 0, input strictly passive (ISP) when δi > 0, output strictly passive (OSP) when δo > 0, and strictly passive (SP) when δi , δo > 0. Let us suppose here, for simplicity, the scalar case n3 = 1 and P33 = 1. The multivariable case requires a more careful treatment and will be reported elsewhere. First, the L2 norm of the output is
kR yk2 =
Z
(Σk δ (t − tk )|y(tk )|sign (y(t )))2 dt ,
and after some manipulation kR yk2 = (1/) Σk |y(tk )|2 . Then, the inner product hv, yi = hR y, yi is
hR y, yi =
Z
(Σk δ (t − tk )|y(tk )|sign (y(t ))) y(t )dt , Rt
+
so that hR y, yi = (1/)Σk |y(tk )| t k |y(t )| dt. Now, since y is k a filtered L2 signal, that is y ∈ FL?2e , |˙y| is bounded; then one can
Rt
+
obtain t k |y(t )|dt ≥ α() |y(tk )|, where the coefficient α() > k 0 can be obtained from (3.26), R by taking the worst-case evolution of y(t ) (least integral value |y|), as a function of the corresponding bounds |yk | ≥ ymin and |˙yX | ≤ (kHAk + kHAd k) Xmax . It also becomes clear that when → 0 the evolution for (t , yR(t )) in (3.26) is a straight line from (tk , yk ) to (tk + , 0), so that |y| = yk /2, meaning that α() → 1/2 when is small enough. Finally, comparing hR y, yi and kR yk2 ,
hR y, yi ≥ α Σk |y(tk )|2 = αkR yk2 ≥ δkR yk2 , and thus, from (3.27), R is output strictly passive for some δ such that 0 < δ ≤ α . Remark 3. An important consequence of the last result is that the stability condition for the feedback system {BLSio , −R } can be translated into a frequencial condition on BLSio . To see this, recall that a negative feedback system {G1 , −G2 } is finite-gain stable if any of (i)–(iii) holds (Carrasco, Baños, & van der Schaft, 2008): (i) G1 , G2 are ISP, (ii) G1 , G2 are OSP, (iii) G1 is SP and G2 is passive.
Clearly, we are interested in case (ii), so if BLSio is OSP, then closed loop stability (finite gain) is guaranteed. Recall also that (Carrasco et al., 2008) a stable and controllable LTI system G(s) is OSP if there exists some δo > 0 such that G(jω) + G? (jω) ≥ δo kG(jω)k2 . In the scalar case, this means that G(jω) lies inside a circle with center s = 1/(2δo ) and radius r = 1/(2δo ). In summary, the required stability condition is Hβ (jω) + Hβ? (jω) ≥
δkHβ (jω)k2 , δ > 0; that is, an OSP condition on −1 B. Hβ (jω) = H jωI − A − Ad e−jωh
(3.28)
In this way, this is a generalization of the results in Beker et al. (2004). In the particular case whenAd → 0 (without time delay), and when H = Mβ C1 , Mβ C2 , P33 → Mβ C1 , O, P33 , one gets the so-called Hβ SPR (positive realness) condition in Beker et al. (2004). Strictly speaking, this passivity analysis ensures stability of the loop {Hβ (s), −R } in the I/O sense. In general, recovering internal stability would require mild assumptions (controllability, detectability, . . . ) However, the passivity result here is presented as an extra interpretation of an original LK analysis which directly ensures internal asymptotic stability. Remark 4 (Robustness in Terms of Small-Gain and Positive Uncertainties). One nice feature of the LK functional (3.1) (Park, 1999) is that it provides an interpretation within a scaled small-gain framework (Zhang et al., 2001). Similarly, the previous passivity analysis can be recast in a robust stability setting. These two results could be combined, providing explicitly which form of uncertainty can be tolerated. It is not included, for brevity, but it is another advantage of the proposed approach. Remark 5 (Comparison with the KYP Approach). One standard tool for passing from time-domain LMIs to frequency-domain conditions is the Kalman–Yacubovitch–Popov (KYP) lemma, used in Baños and Barreiro (2009). The KYP lemma here gives rise to big matrix conditions, and is a more involved approach. 4. Example Consider a feedback system according to the setup of Fig. 1. Here the plant has a transfer function e−hs P0 (s) given by (the example is adapted from Beker et al. (2004)) s+1 , (4.1) s(s + 0.2) and the feedback compensator K is a FORE compensator with base LTI system given by Kbl (s) = 1/(s + 1). It can be shown that the base LTI control system is stable for delays h < 0.2. In the following, the passivity result in Section 3 is used to show that the reset action does not destabilize the system, by checking that (3.28) is OSP. A delay h = 0.15 is chosen. In this simple case the plant has two states and the reset compensator has one state only; thus n1 = 2, n2 = 0, and n3 = 1. Let us use the realizations of P0 (s) and Kbl (s): e−hs P0 (s) = e−hs
Ap =
−0.2 1
0 , 0
Bp =
1 , 0
Cp = 1
1
(4.2)
and Ac = −1, Bc = 1, Cc = 1. Here Mβ = (β) and P33 are scalar, and P33 can be chosen as P33 = 1 without loss of generality. As a result, matrices H and B are given by (3.14): H = Mβ Cp
P33 = β
β
1 ,
B> = 0
0
1 .
(4.3)
With the above values, system (3.28) is OSP for values of β between 0.29 and 0.56. As a result, the stability of the reset control system is guaranteed, in this case for h = 0.15. An identical procedure shows that the reset action never destabilizes a stable base LTI control system and its stability limit is h = 0.2, the same limit as the base LTI control system. Fig. 2 plots the time response, showing the typical fast transient and not excessive overshoot gained with reset.
A. Barreiro, A. Baños / Automatica 46 (2010) 216–221
Fig. 2. Time response yp (t ) (top) and yc (t ) (bottom).
5. Conclusions The stability analysis developed in this work is based on the Lyapunov functional V (xt ) (3.1), for which it is required (Proposition 3.1) that (i) V˙ (xt ) < 0 and that (ii) 1V (xt ) ≤ 0 at the reset surface. For the first condition, (i) V˙ (xt ) < 0, to avoid conservativeness, we have used a quite general functional V (xt ) (3.1) containing several free matrix parameters, with good accuracy and low computational cost. For the second condition, (ii) 1V (xt ) ≤ 0, we have contributed with the passivity analysis. The main idea is the interpretation of the overall system as the negative feedback of the base system BLSio with the impulsive resetting subsystem R. This interpretation puts the problem in a well-known framework and facilitates further treatment using frequency domain results. The case when the base system is unstable and resetting is used to stabilize it is not treated in this paper, and it cannot be treated using Proposition 3.1. It requires allowing V˙ (xt ) or 1V (xt ) to be positive sometimes, provided that, on the average, V decreases. A good idea is to bound the time intervals between resetting instants and use a discrete-time analysis. This strategy was applied in Baños et al. (2007) to unstable plants without delay. The extension to delayed cases is an interesting open problem. Finally, Carrasco et al. (2008) deals with the stability of nonlinear reset systems, using passivity and dissipativity techniques. One of the application areas where time delays are more critical is the field of teleoperation. In Fernández, Barreiro, Baños, and Carrasco (2008), the reset strategy is applied to simulated teleoperation systems. Achieving an actual performance improvement in real teleoperated systems in one of the most promising advantages of reset control of time-delay systems. References Åström, K. J. (2000). Limitations on control system performance. European Journal of Control, 6, 2–20. Bainov, D. D., & Simeonov, (1989). Systems with impulse effect: Stability, theory and applications. Chichester: Ellis Horwood. Baños, A., & Barreiro, A. (2009). Delay-independent stability of reset systems. IEEE Transactions on Automatic Control, 341–346. Baños, A., Carrasco, J., & Barreiro, A. (2007). Reset-times dependent stability of reset control with unstable base systems. In Proceedings of the IEEE Int. Symp. on Industrial Electronics (ISIE), Vigo, Spain, June 4-7. Beker, O, Hollot, C. V., Chait, Y., & Han, H. (2004). Fundamental properties of reset control systems. Automatica, 40, 905–915.
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Antonio Barreiro was born in Vigo, Spain in 1959. He received his degrees of Ingeniero and Doctor Ingeniero Industrial from the Polytechnic University of Madrid (UPM) in 1984 and 1989, respectively. From 1984 to 1987 he was with the Departamento de Matemática Aplicada of the E.T.S. Ingenieros Industriales of the UPM. Since 1987 he has been with the Departamento de Ingeniería de Sistemas y Automática of the University of Vigo, where he is now Professor in the area of Automatic Control. His research interests include nonlinear and robust stability, time-delay systems and reset control, with applications in networked control and teleoperation systems.
Alfonso Baños was born in Córdoba, Spain in 1965. He received his degrees of Licenciado and Doctor in Physics from the University of Madrid (Complutense) in 1987 and 1991, respectively. From 1988 to 1992 he was with the Instituto de Automática Industrial (C.S.I.C.) in Madrid, where he pursued research in nonlinear control and robotics. In 1992 he joined the University of Murcia, where he is currently Professor in the area of Automatic Control. He has also held visiting appointments at the University of Strathclyde, the University of Minnesota at Minneapolis, and the University of California at Berkeley. His research interests include robust and nonlinear control, and reset control, with applications in process control and networked control systems.